Ten-moment two-fluid extended MHD

Equations for a two-fluid ten-moment extended MHD model (see discussion of naming plasma models below) are:

continuity:

    n,t + ∇•(n*u) = 0;

    where
      n = ni = ne = particle density and
      u = bulk fluid velocity,

conservation of momentum:

    (ρ*u),t + ∇•(ρ*u⊗u+P) = J × B

  or

    (ρ*u),t + ∇•(ρ*u⊗u+P) + ∇•(I⊗B•B/2-B⊗B)/μ0 = 0;

  where
    ρ = m*n is mass density,
    m = mi + me is sum of particle masses,
    P = Pi + Pe is total pressure tensor,
    B = magnetic field,
    μ0 = magnetic permeability,

pressure tensor evolution for each species:

  Ps,t + ∇•(us⊗Ps) + 2*Sym(Ps•∇ us) + ∇•(heat_fluxs)
    = (qs/ms)Sym(Ps × B) + (ps*I - Ps)/τs,
      + resistive_heatings + thermal_equilibrations

  where
    ,t denotes partial derivative with respect to time,
    * denotes (possibly tensor) product,
    ⊗ denotes tensor product,
    × denotes cross product,
    Sym denotes symmetrization of its argument tensor,
    I is the identity tensor,
    Ps is the species pressure tensor
    ms is particle mass,
    us := u + ws  is species bulk velocity,
    ws = J*μ/(ms*qs*n)  is species drift velocity,
    qs = is particle charge,
    qi = e is charge on a proton,
    qe = -e is charge on an electron,
    n = ni = ne  is species particle density,
    J = ∇×(B)/μ0  is the current according to Ampere's law,
    τs = τ0*sqrt(ms))*Ts(3/2)ns
      is a Braginskii-type closure for the isotropization period,
    μs =  τs* ps is the viscosity
      (in the five-moment Hall MHD model, which asymptotically agrees with the
      ten-moment model on time scales longer than the isotropization period),
    τ0 = 3*(2*π)(3/2)*(ε0/e2)2/ln(Λ),
    ε0 = permittivity of free space,
    ln(Λ) = Coulomb logarithm,
    Ts = ps/ns is the temperature,
    ps = trace(Ps)/3 is the scalar pressure,
    resistive_heatingi
      = 2*(mi/m)*η*n2*wi•(a//-a⊥)I⊗wi+a⊥*wi⊗I)
      (where a// and a⊥ specify allocation of frictional heating
        among parallel and perpendicular modes ...[not done]),
    thermal_equilibrationi = K*n2(Te-Ti),
    thermal_equilibratione = -thermal_equilibrationi,
      (where K is ...[not done]),
    Ts = Ps/ns is temperature tensor,
    heat_fluxs is the species heat flux tensor (third-order, see below);

induction equation:

  B,t + ∇×(E) = 0,

    where the electric field E is given by

Ohm's law:

  E = η*J + B×u
      + hallTerm
      + pressureTerm
      + inertialTerm
  where
    η = resistivity = μ/(e2n*τslowing), where
      τslowing = .51*τ0*(Ti/mi+Te/me)(3/2)*μ2/n,
    hallTerm = J×B*dm/(m*e*n),
    pressureTerm = ∇•(me*Pi - mi*Pe)/(m*e*n),
    inertialTerm = (J,t+∇•(u⊗J + J⊗u - J⊗J*dm/(m*e*n)))*μ/(e2*n),

    where
      m = mi + me = total particle mass,
      dm = mi - me = particle mass difference,
      μ = mi*me/m = reduced particle mass.

    Alternate expressions manifestly symmetric among species are

      -dm/(m*e) = μ/(qi*mi) + μ/(qe*me)
      

An entropy-respecting closure for the third-order heat flux tensor is

  heat_fluxs = a1*3*Sym(∇ (Ts-1)),

    where
      Ts-1 = inverse of temperature tensor,
      a1= (2/5)κs*Ts2,
      κs = species heat conductivity, and the Braginskii closure gives
      κs = 3.91*ps*τs/ms.
    
    Note that for heat flux to preserve positivity the temperature used in
      the Braginskii relations for τs and a1 should be taken as

    Ts := 3√(det( Ts))

      (the geometric average of the eigenvalues of the temperature tensor)
      rather than Ts = (tr( Ts))/3
      (the arithmetic average of the eigenvalues of the temperature tensor)

  Using the 5-moment heat flux instead of heat_flux can
  violate entropy and can make the pressure tensor cease to be
  positive-definite.